An extension of Paulsen–Gjessing’s risk model with stochastic return on investments

We consider in this paper a general two-sided jump-diffusion risk model that allows for risky investments as well as for correlation between the two Brownian motions driving insurance risk and investment return. We first introduce the model and then find the integro-differential equations satisfied by the Gerber–Shiu functions as well as the expected discounted penalty functions at ruin caused by a claim or by oscillation. We also study the dividend problem for the threshold and barrier strategies, the moments and moment-generating function of the total discounted dividends until ruin are discussed. Some examples are given for special cases.